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Zhong, J. Monte Carlo Simulations. Encyclopedia. Available online: https://encyclopedia.pub/entry/60634 (accessed on 07 November 2026).
Zhong J. Monte Carlo Simulations. Encyclopedia. Available at: https://encyclopedia.pub/entry/60634. Accessed November 07, 2026.
Zhong, Jack. "Monte Carlo Simulations" Encyclopedia, https://encyclopedia.pub/entry/60634 (accessed November 07, 2026).
Zhong, J. (2026, October 08). Monte Carlo Simulations. In Encyclopedia. https://encyclopedia.pub/entry/60634
Zhong, Jack. "Monte Carlo Simulations." Encyclopedia. Web. 08 October, 2026.

Monte Carlo Simulations

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Monte Carlo simulations are computational experiments in which a system is sampled by repeated random draws from a probability distribution instead of being solved deterministically. In statistical physics the aim is usually to evaluate an average over an ensemble whose state space is too large to enumerate, and the method replaces that enumeration by a stochastic walk that visits the important states in proportion to their Boltzmann weight; its usefulness therefore rests on ergodic sampling and on the central limit theorem rather than on any physical approximation [1]. Quantum lattice models were among the first systems treated this way, where the mapping to a classical problem in one higher dimension makes the sampling tractable [2]. The principal limitation is that a local update moves slowly through a configuration space with high barriers, so non-local moves that update a whole cluster or a whole collective coordinate at once were developed to overcome critical slowing down [3]. The same sampling idea solves kinetic and transport equations by tracking representative particles rather than a grid [4]. Because every result carries a statistical error, that error has to be estimated from the run itself, and its analysis is a standard part of the technique [5].

stochastic sampling random walk statistical physics markov chain error estimation computational physics
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References

  1. Meshkov, S.V. Simulation by the Monte Carlo method in statistical physics. Soviet Physics Uspekhi 1989, 32, 833-833. [CrossRef]
  2. De Raedt, H.; Lagendijk, A. Monte Carlo simulation of quantum statistical lattice models. Physics Reports 1985, 127, 233-307. [CrossRef]
  3. Janke, W. Nonlocal Monte Carlo algorithms for statistical physics applications. Mathematics and Computers in Simulation 1998, 47, 329-346. [CrossRef]
  4. Babovsky, H. Monte Carlo simulation schemes for steady kinetic equations. Transport Theory and Statistical Physics 1994, 23, 249-264. [CrossRef]
  5. Chen, G.; Boyd, I.D. Statistical Error Analysis for the Direct Simulation Monte Carlo Technique. Journal of Computational Physics 1996, 126, 434-448. [CrossRef]
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